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  <h1>Source code for fatghol.valences</h1><div class="highlight"><pre>
<span class="c">#! /usr/bin/env python</span>
<span class="c">#</span>
<span class="sd">&quot;&quot;&quot;Functions related to valences of ribbon graphs.</span>
<span class="sd">&quot;&quot;&quot;</span>
<span class="c">#</span>
<span class="c">#   Copyright (C) 2008-2012 Riccardo Murri &lt;riccardo.murri@gmail.com&gt;</span>
<span class="c">#   All rights reserved.</span>
<span class="c">#</span>
<span class="c">#   This program is free software: you can redistribute it and/or modify</span>
<span class="c">#   it under the terms of the GNU General Public License as published by</span>
<span class="c">#   the Free Software Foundation, either version 3 of the License, or</span>
<span class="c">#   (at your option) any later version.</span>
<span class="c">#</span>
<span class="c">#   This program is distributed in the hope that it will be useful,</span>
<span class="c">#   but WITHOUT ANY WARRANTY; without even the implied warranty of</span>
<span class="c">#   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the</span>
<span class="c">#   GNU General Public License for more details.</span>
<span class="c">#</span>
<span class="c">#   You should have received a copy of the GNU General Public License</span>
<span class="c">#   along with this program.  If not, see &lt;http://www.gnu.org/licenses/&gt;.</span>
<span class="c">#</span>
<span class="n">__docformat__</span> <span class="o">=</span> <span class="s">&#39;reStructuredText&#39;</span>


<span class="c">#import cython</span>


<span class="kn">from</span> <span class="nn">fatghol.combinatorics</span> <span class="kn">import</span> <span class="n">PartitionIterator</span>


<span class="c">#@cython.ccall</span>
<span class="c">#@cython.locals(g=cython.int, n=cython.int,</span>
<span class="c">#               L=cython.int, V=cython.int, result=set)</span>
<div class="viewcode-block" id="vertex_valences_for_given_g_and_n"><a class="viewcode-back" href="../../api.html#fatghol.valences.vertex_valences_for_given_g_and_n">[docs]</a><span class="k">def</span> <span class="nf">vertex_valences_for_given_g_and_n</span><span class="p">(</span><span class="n">g</span><span class="p">,</span><span class="n">n</span><span class="p">):</span>
    <span class="sd">&quot;&quot;&quot;Return all possible valences of fatgraphs appearing in the</span>
<span class="sd">    `(g,n)` complex.</span>

<span class="sd">    The returned partitions are sorted in ascending order.</span>

<span class="sd">    Examples::</span>
<span class="sd">      &gt;&gt;&gt; vertex_valences_for_given_g_and_n(0,3)</span>
<span class="sd">      [(3, 3), (4,)]</span>
<span class="sd">      &gt;&gt;&gt; vertex_valences_for_given_g_and_n(1,1)</span>
<span class="sd">      [(3, 3), (4,)]</span>
<span class="sd">    &quot;&quot;&quot;</span>
    <span class="c"># with 1 vertex only, there are this many edges:</span>
    <span class="n">L</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">g</span> <span class="o">+</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
    <span class="n">V</span> <span class="o">=</span> <span class="mi">1</span>
    <span class="n">result</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">PartitionIterator</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">L</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
    <span class="k">while</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">L</span> <span class="o">&gt;=</span> <span class="mi">3</span><span class="o">*</span><span class="n">V</span><span class="p">):</span>
        <span class="n">V</span> <span class="o">+=</span> <span class="mi">1</span>
        <span class="n">L</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">g</span> <span class="o">+</span> <span class="n">n</span> <span class="o">+</span> <span class="n">V</span> <span class="o">-</span> <span class="mi">2</span> <span class="c"># by Euler&#39;s formula</span>
        <span class="c">## Each edge has *two* vertex endpoints (possibly the same vertex),</span>
        <span class="c">## so the set of vertex valences is a partition of `2*L`, where `L`</span>
        <span class="c">## is the number of edges, with some further constraints:</span>
        <span class="c">##   - each vertex has valence at least 3;</span>
        <span class="c">##   - thus, no vertex can have valence greater than `2*L-3*(V-1)`,</span>
        <span class="c">##     that is, supposing all vertices except one are 3-valent,</span>
        <span class="c">##     the highest-valent vertex must have valence `2L - 3(V-1)`.</span>
        <span class="n">result</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">PartitionIterator</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">L</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">L</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="p">(</span><span class="n">V</span><span class="o">-</span><span class="mi">1</span><span class="p">))))</span>
    <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>





<span class="c">## main: run tests</span>
</div>
<span class="k">if</span> <span class="s">&quot;__main__&quot;</span> <span class="o">==</span> <span class="n">__name__</span><span class="p">:</span>
    <span class="kn">import</span> <span class="nn">doctest</span>
    <span class="n">doctest</span><span class="o">.</span><span class="n">testmod</span><span class="p">(</span><span class="n">optionflags</span><span class="o">=</span><span class="n">doctest</span><span class="o">.</span><span class="n">NORMALIZE_WHITESPACE</span><span class="p">)</span>
</pre></div>

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